Recently, I was reading

*Young Children Reinvent Arithmetic: Implications of Piaget's Theory*by Constance Kamii and came across one of her games that she plays with first graders in her game-driven curriculum.

The game was called double war. Each child got two piles. Each turn consisted of both players turning over a card from each of their piles for a total of two cards each. They would then decide who had the highest sum. This game--although powerful for first graders--holds little value for my high school students. I got to thinking how I could capture the spirit of the game as well as a relevance for the grade nine and ten range.

I wanted to stick with the "war" theme. I also wanted students to decide whether numbers were larger or smaller. I also thought I would target fractions because they present my students with perennial difficulty. The bulk of the task would be completed in pairs.

I strategically chose a series of numbers between one and ten. The goal of the task is for the students to create an "army" of fractions (so to speak). The army needs to consist of the largest ten fractions possible. (they are given 20 numbers total). That goal is designed to create discourse within and between groups.

What makes an army the largest? The sum of the entire army is the largest possible? Maybe two armies are compared on a fraction-by-fraction basis? Each army could send out their champion and they could battle.

To further increase discourse and challenge, I added two conditions to the army creation:

- No two fractions can be the same
- You can not create a fraction that will reduce

The task is delivered to the pairs in three stages. Each stage has an accompanying handout. For convenience, I have linked this post to each handout as a Google Doc.

Stage 1 - Entry

Each group gets a copy of the Playing Cards. The handout includes two copies of the twenty numbers. In stage one, I want students to begin to discuss how they will approach the problem. Nothing is formally recorded, but I am sure to stop by and pick the brains of each pair. Even in stage one I am hoping for the discussion to bleed between groups.

Stage 2 - Formalization

I give each group a copy of the How To... handout where they are required to talk through their strategy to create the strongest set of ten fractions. I really encourage students to work with the restraints that they are given. Mathematics is about sense making around a set of parameters. Here, students should use the fact that no two fractions can be identical as proof; relatively prime numbers should become a logical truth in their arsenal. This stage usually lasts fairly long. I am sure to gather main ideas from groups as they write. These carefully selected and sequenced ideas will become the basis of a class discussion.

Stage 3 - Extension

After a class discussion, I give each pair an Army to Beat handout. If your class is short, this may end up being a take home assignment. It would be a great one for students to pull out after dinner with parents. Here, students are asked to design an army that "beats" a carefully pre-designed army. Again, their interpretation of "beat" is important. Some may match up fractions mano-a-mano while others will justify based on a global sum. Both hold important mathematics.

The goal of the task is to get students thinking about how fractions work and what they mean. I want them to get a deep understanding of what happens to fractions as their numerators get smaller or their denominators get larger (or vice versa). I want them to realize that fractions are a composition of numbers and they have a spot on the number line like other numbers. All of these deeper, more robust learnings are bolstered by the atomic skills of building, comparing, and adding fractions. Hopefully these things are all captured within Kamii's framework of autonomy provided through self-directed games in math class.

NatBanting